Problem: All of the 5th grade teachers and students from Loyola went on a field trip to an archaeology museum. Tickets were $$7.00$ each for teachers and $$3.50$ each for students, and the group paid $$52.50$ in total. The next month, the same group visited a science museum where the tickets cost $$28.00$ each for teachers and $$7.50$ each for students, and the group paid $$151.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+3.5y = 52.5}$ ${28x+7.5y = 151.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-28x-14y = -210}$ ${28x+7.5y = 151.5}$ Add the top and bottom equations together. $ -6.5y = -58.5 $ $ y = \dfrac{-58.5}{-6.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {7x+3.5y = 52.5}$ to find $x$ ${7x + 3.5}{(9)}{= 52.5}$ $7x+31.5 = 52.5$ $7x = 21$ $x = \dfrac{21}{7}$ ${x = 3}$ You can also plug ${y = 9}$ into $ {28x+7.5y = 151.5}$ and get the same answer for $x$ ${28x + 7.5}{(9)}{= 151.5}$ ${x = 3}$ There were $3$ teachers and $9$ students on the field trips.